Effect of Tendon-Related Variables on the Behavior of Externally CFRP Prestressed Concrete Beams

This work assesses the flexural performance of prestressed concrete beams with external carbon fiber-reinforced polymer (CFRP) tendons, focusing on tendon-related variables. A finite element analysis (FEA) method is verified. A numerical parametric analysis of prestressed concrete beams with external CFRP tendons is carried out. Four tendon-related variables are considered, namely, the area, initial prestress, depth and elastic modulus of tendons. The analysis shows that flexural ductility decreases as the tendon area, initial prestress or elastic modulus increases but is insensitive to the tendon depth. The ultimate tendon stress increment (Δσp) is influenced by all of the four variables investigated. JGJ 92-2016 (Chinese technical specification for concrete structures prestressed with unbonded tendons) significantly underestimates Δσp and, hence, is over-conservative for the strength design of these beams. An equation is proposed for calculating Δσp, taking into account all four variables investigated. An analytical model is then developed to estimate the flexural strength (Mu) of prestressed concrete beams with external CFRP tendons. The proposed analytical model shows good agreement with FEA, i.e., the mean discrepancy for Δσp is 0.9% with a standard deviation of 11.1%; and the mean discrepancy for Mu is −1.6% with a standard deviation of 2.1%.


Introduction
External prestressing is a powerful technique for structural rehabilitation and construction [1,2]. The major concern for this technique is the corrosive damage of external tendons made of conventional prestressing steel. The utilization of fiber-reinforced polymer (FRP) reinforcement in engineering is widespread [3,4]. The composite reinforcement may be made of glass fiber-reinforced polymer (GFRP) [5], aramid fiber-reinforced polymer (AFRP) [6], basalt fiber-reinforced polymer (BFRP) [7] or carbon fiber-reinforced polymer (CFRP) [8]. Of various FRP composites, carbon fiber-reinforced polymer (CFRP) possesses the best resistance to creep, namely, it can sustain around 80% of the tensile strength without experiencing creep rupture [9,10]. Therefore, CFRP is recognized to be ideal for prestressing applications [11][12][13]. This non-corrosive composite material is particularly suitable for external tendons in lieu of conventional prestressing steel in overcoming the corrosive problem [14,15], as these tendons are exposed to harsh conditions.
Researchers have paid much attention to prestressed concrete beams with external CFRP tendons. Grace et al. [16] performed a set of laboratory tests including one reference box-specimen without external tendons, one box-specimen prestressed by external CFRP tendons and one box-specimen with external CFRP tendons without prestressing. Their tests showed that the prestressing of girders by external CFRP tendons significantly Materials 2023, 16, 5197 3 of 16 It is assumed that the stress-strain (σ s -ε s ) law for steel bars [10] is represented by σ s = E s ε s for ε s ≤ ε y 0 for ε s > ε y (4) where E s is the steel elastic modulus and ε y is the yield strain. An FEA method for prestressed beams with external tendons was developed [23]. The following assumptions are adopted: (1) plane section remaining plane (excluding external tendons due to their unbonded nature [24]); (2) negligible bond-slip for internal steel bars; (3) negligible shear deformation (given that prestressed concrete beams generally have a high slenderness); (4) negligible frictional loss between external tendons and deviators (this assumption is reasonable in practice but may lead to a bit of an overestimation of the tendon stress). The Euler-Bernoulli beam theory is employed. The material laws for concrete and steel bars are introduced by applying the layered method. External prestressing is transformed into equivalent loads. The analysis consists of two steps. The first step is to determine the stress and deformation at self-weight only. The second step is to perform the full-range analysis from zero loads up to failure. During the analysis, as long as concrete crushing or reinforcement rupture occurs, the beam fails. Details of the FEA method were reported in [23].

Verification
Two beam specimens (B4 and B5) are selected herein [17]. The T-beams had a clear span of 3000 mm, subjected to two concentrated loads at third points, as illustrated in Figure 1. External tendons were attached with one deviator at the midspan. The specimens were designed to be identical except for the prestress level, i.e., the initial prestress was 396 MPa for B4 and 889 MPa for B5. CFRP tendons had an area of 100. 5  It is assumed that the stress-strain (σs-εs) law for steel bars [10] (4) where Es is the steel elastic modulus and εy is the yield strain.
An FEA method for prestressed beams with external tendons was developed [23]. The following assumptions are adopted: (1) plane section remaining plane (excluding external tendons due to their unbonded nature [24]); (2) negligible bond-slip for internal steel bars; (3) negligible shear deformation (given that prestressed concrete beams generally have a high slenderness); (4) negligible frictional loss between external tendons and deviators (this assumption is reasonable in practice but may lead to a bit of an overestimation of the tendon stress). The Euler-Bernoulli beam theory is employed. The material laws for concrete and steel bars are introduced by applying the layered method. External prestressing is transformed into equivalent loads. The analysis consists of two steps. The first step is to determine the stress and deformation at self-weight only. The second step is to perform the full-range analysis from zero loads up to failure. During the analysis, as long as concrete crushing or reinforcement rupture occurs, the beam fails. Details of the FEA method were reported in [23].

Verification
Two beam specimens (B4 and B5) are selected herein [17]. The T-beams had a clear span of 3000 mm, subjected to two concentrated loads at third points, as illustrated in Figure 1. External tendons were attached with one deviator at the midspan. The specimens were designed to be identical except for the prestress level, i.e., the initial prestress was 396 MPa for B4 and 889 MPa for B5. CFRP tendons had an area of 100.  A comparison between the test and FEA results regarding the load versus deflection and load versus tendon force for two specimens is illustrated in Figure 2. It is seen that the FEA is able to reasonably capture the key response characteristics of the specimens during the complete loading stages, including the cracking, yielding and ultimate limit states. In addition, both test and FEA results demonstrate that Specimen B5 exhibits higher cracking and ultimate loads but a lower ultimate deflection and an increase in the tendon force compared to Specimen B4. This observation indicates the importance of the prestress A comparison between the test and FEA results regarding the load versus deflection and load versus tendon force for two specimens is illustrated in Figure 2. It is seen that the FEA is able to reasonably capture the key response characteristics of the specimens during the complete loading stages, including the cracking, yielding and ultimate limit states. In addition, both test and FEA results demonstrate that Specimen B5 exhibits higher cracking and ultimate loads but a lower ultimate deflection and an increase in the tendon force compared to Specimen B4. This observation indicates the importance of the prestress level. In the following section, this variable is analyzed in more detail along with other important variables related to CFRP tendons (tendon area, depth and elastic modulus) level. In the following section, this variable is analyzed in more detail along with other important variables related to CFRP tendons (tendon area, depth and elastic modulus)  Figure 3 shows an externally prestressed concrete reference beam for numerical parametric study. The fck is 60 MPa. The tendon area is 1100 mm 2 . CFRP tendons have a rupture strength of 1840 MPa, an elastic modulus of 150 GPa and an initial prestress of 1104 MPa. Either the bottom or top steel bars have a cross-sectional area of 360 mm 2 , yield strength of 450 MPa and elastic modulus of 200 GPa.   level. In the following section, this variable is analyzed in more detail along with other important variables related to CFRP tendons (tendon area, depth and elastic modulus)

Effect of Tendon Area
The tendon area, A p , varies between 200 and 2000 mm 2 to examine its effect on the structural response. Figure 4a shows that, at a small area of 200 mm 2 , the prestressing effect is not prominent, just counteracting the effect of the self-weight, i.e., the initial deflection prior to the live load is around zero. At A p = 650 mm 2 or above, the prestressing effect becomes pronounced, leading to an upward deflection at the initial state. However, at high areas of 1550 and 2000 mm 2 , there occur excessive prestressing effects. In these cases, the top face of the critical section has cracked prior to load application, and the upward deflection is rather big (13.5 and 18.3 mm). It is observed in Figure 4a that as the tendon area increases, the cracking and ultimate loads significantly increase, while the ultimate deflection obviously decreases. In this analysis, increasing the tendon area from 200 to 2000 mm 2 leads to an increase in the ultimate load by 528.5% and a decrease in the ultimate deflection by 18.3%. Figure 4b shows that, at the initial state, the self-weight moment is 56 kN·m. A higher tendon area causes substantially higher flexural stiffness and flexural strength but a markedly smaller ultimate curvature.

Effect of Tendon Area
The tendon area, Ap, varies between 200 and 2000 mm 2 to examine its effect on the structural response. Figure 4a shows that, at a small area of 200 mm 2 , the prestressing effect is not prominent, just counteracting the effect of the self-weight, i.e., the initial deflection prior to the live load is around zero. At Ap = 650 mm 2 or above, the prestressing effect becomes pronounced, leading to an upward deflection at the initial state. However, at high areas of 1550 and 2000 mm 2 , there occur excessive prestressing effects. In these cases, the top face of the critical section has cracked prior to load application, and the upward deflection is rather big (13.5 and 18.3 mm). It is observed in Figure 4a that as the tendon area increases, the cracking and ultimate loads significantly increase, while the ultimate deflection obviously decreases. In this analysis, increasing the tendon area from 200 to 2000 mm 2 leads to an increase in the ultimate load by 528.5% and a decrease in the ultimate deflection by 18.3%. Figure 4b shows that, at the initial state, the self-weight moment is 56 kN·m. A higher tendon area causes substantially higher flexural stiffness and flexural strength but a markedly smaller ultimate curvature. The flexural ductility can be expressed by either deflection or curvature ductility, defined as the ratio of deflection or curvature at ultimate to that at yielding. Table 1 presents the data for the flexural ductility of beams with different tendon areas. It is seen that the flexural ductility quickly deceases as the tendon area increases, i.e., increasing the tendon area from 200 to 2000 mm 2 leads to a decrease in deflection ductility by 70.8% and curvature ductility by 73.9%.  Figure 5a,b illustrate the tendon stress versus midspan moment and deflection for beams with various tendon areas, respectively. In the initial range of loading up to tensile cracking, the rate of increase in tendon stress with the moment is independent of the tendon area, as demonstrated in Figure 5a. A higher tendon area causes a higher cracking moment and, correspondingly, a larger increase in tendon stress at cracking. Moreover, The flexural ductility can be expressed by either deflection or curvature ductility, defined as the ratio of deflection or curvature at ultimate to that at yielding. Table 1 presents the data for the flexural ductility of beams with different tendon areas. It is seen that the flexural ductility quickly deceases as the tendon area increases, i.e., increasing the tendon area from 200 to 2000 mm 2 leads to a decrease in deflection ductility by 70.8% and curvature ductility by 73.9%.  Figure 5a,b illustrate the tendon stress versus midspan moment and deflection for beams with various tendon areas, respectively. In the initial range of loading up to tensile cracking, the rate of increase in tendon stress with the moment is independent of the tendon area, as demonstrated in Figure 5a. A higher tendon area causes a higher cracking moment and, correspondingly, a larger increase in tendon stress at cracking. Moreover, the ultimate stress increase in tendons decreases as the tendon area increases. In this analysis, increasing the tendon area from 200 to 2000 mm 2 reduces the ultimate stress increase in external tendons by 24.4%. Figure 5b demonstrates that, for the same deflection, a high tendon area causes a small increase in tendon stress.

Effect of Prestress Level
Five initial prestress levels are considered, i.e., 0, 20, 40, 60 and 80% of the rupture strength, or, equivalently, the initial prestress, σp0, corresponds to 0, 368, 736, 1104 and 1472 MPa. Figure 6a,b show the load-deflection and moment-curvature behavior of beams with various prestress levels. The initial deflection is highly dependent on the prestress level, i.e., a higher prestress level corresponds to a larger upward deflection. At a prestress level of zero, there is a downward deflection due to the self-weight effect, as expected. In addition, a higher prestress level effectively improves not only the cracking load and moment but also the ultimate load and flexural strength. It is worth mentioning that this observation is different

Effect of Prestress Level
Five initial prestress levels are considered, i.e., 0, 20, 40, 60 and 80% of the rupture strength, or, equivalently, the initial prestress, σp0, corresponds to 0, 368, 736, 1104 and 1472 MPa. Figure 6a,b show the load-deflection and moment-curvature behavior of beams with various prestress levels. The initial deflection is highly dependent on the prestress level, i.e., a higher prestress level corresponds to a larger upward deflection. At a prestress level of zero, there is a downward deflection due to the self-weight effect, as expected. In addition, a higher prestress level effectively improves not only the cracking load and moment but also the ultimate load and flexural strength. It is worth mentioning that this observation is different The initial deflection is highly dependent on the prestress level, i.e., a higher prestress level corresponds to a larger upward deflection. At a prestress level of zero, there is a downward deflection due to the self-weight effect, as expected. In addition, a higher prestress level effectively improves not only the cracking load and moment but also the ultimate load and flexural strength. It is worth mentioning that this observation is different from the case using bonded tendons. Generally, bonded FRP tendons rupture at failure, and therefore, the prestress level has practically no influence on the ultimate load or flexural strength of the beams [25]. On the other hand, the stress increase in external tendons is much slower than that in bonded tendons due to strain incompatibility between external tendons and the adjacent concrete and also the second-order effects [26]. Therefore, the ultimate stress in external tendons is generally below their rupture strength. The ultimate load or flexural strength is controlled by the tendon stress and is therefore strongly dependent on the prestress level in external tendons. Moreover, a higher prestress level results in a substantially higher flexural stiffness and lower ultimate curvature, while this variable appears to have a marginal effect on the ultimate deflection. Table 2 presents the data for flexural ductility for beams with different prestress levels. It is seen that as the prestress level increases, the flexural ductility quickly decreases. Increasing the initial prestress from 0 to 1472 MPa leads to a decrease in deflection ductility by 70.3% and in curvature ductility by 60.9%.  Figure 7a,b illustrate the effect of the initial prestress on the tendon stress versus midspan moment and deflection, respectively. Over the elastic range, the initial prestress has no impact on the increase in tendon stress as the moment develops, as shown in Figure 7a. A higher initial prestress corresponds to a higher increment in tendon stress at cracking but, generally, a lower one at ultimate. In this analysis, increasing the prestress level from 20 to 80% causes a reduction in the ultimate stress increase in external tendons by 10.7%. It is seen in Figure 7b that, when deflecting, the greater the initial prestress, the slower the increase in tendon stress. from the case using bonded tendons. Generally, bonded FRP tendons rupture at failure, and therefore, the prestress level has practically no influence on the ultimate load or flexural strength of the beams [25]. On the other hand, the stress increase in external tendons is much slower than that in bonded tendons due to strain incompatibility between external tendons and the adjacent concrete and also the second-order effects [26]. Therefore, the ultimate stress in external tendons is generally below their rupture strength. The ultimate load or flexural strength is controlled by the tendon stress and is therefore strongly dependent on the prestress level in external tendons. Moreover, a higher prestress level results in a substantially higher flexural stiffness and lower ultimate curvature, while this variable appears to have a marginal effect on the ultimate deflection. Table 2 presents the data for flexural ductility for beams with different prestress levels. It is seen that as the prestress level increases, the flexural ductility quickly decreases. Increasing the initial prestress from 0 to 1472 MPa leads to a decrease in deflection ductility by 70.3% and in curvature ductility by 60.9%.  Figure 7a,b illustrate the effect of the initial prestress on the tendon stress versus midspan moment and deflection, respectively. Over the elastic range, the initial prestress has no impact on the increase in tendon stress as the moment develops, as shown in Figure  7a. A higher initial prestress corresponds to a higher increment in tendon stress at cracking but, generally, a lower one at ultimate. In this analysis, increasing the prestress level from 20 to 80% causes a reduction in the ultimate stress increase in external tendons by 10.7%. It is seen in Figure 7b that, when deflecting, the greater the initial prestress, the slower the increase in tendon stress.

Effect of Tendon Depth
The initial tendon depth (maximum depth), d p , ranging from 400 to 600 mm, is used to investigate its effect on the structural response. The load-deflection and moment-curvature behavior are shown in Figure 8a,b, respectively. From Figure 8, it is seen that a larger tendon depth registers a larger upward deflection prior to load application. In addition, the cracking and ultimate loads or moments substantially increase as the tendon depth increases. In this analysis, increasing the tendon depth from 400 to 600 mm 2 causes an increase in the ultimate load by 74.2%. However, the ultimate deflection, curvatures and flexural ductility are insensitive to the tendon depth, as presented in Table 3.

Effect of Tendon Depth
The initial tendon depth (maximum depth), dp, ranging from 400 to 600 mm, is used to investigate its effect on the structural response. The load-deflection and moment-curvature behavior are shown in Figure 8a,b, respectively. From Figure 8, it is seen that a larger tendon depth registers a larger upward deflection prior to load application. In addition, the cracking and ultimate loads or moments substantially increase as the tendon depth increases. In this analysis, increasing the tendon depth from 400 to 600 mm 2 causes an increase in the ultimate load by 74.2%. However, the ultimate deflection, curvatures and flexural ductility are insensitive to the tendon depth, as presented in Table 3.   Figure 9a,b illustrate the tendon stress versus midspan moment and deflection for beams with various tendon depths, respectively. As the tendon depth increases, the ultimate tendon stress increases substantially. As mentioned previously, the beams with different tendon depths exhibit approximately the same deformation at ultimate, implying there are approximately the same concrete strain distributions along the depth of the sections of these beams. Therefore, the larger the tendon depth, the larger the average change in the concrete strain at the same level of the external tendons and, thereby, the larger the tendon stress increase. Due to a lower reduction in the effective tendon depth, a larger tendon depth assumes relatively less significant second-order effects, which would reduce relatively less tendon stresses. In this analysis, increasing the tendon depth from 400 to 600 mm causes an increase in the ultimate tendon stress increment by 93.2%. For the same deflection, the greater the tendon depth, the greater the tendon stress increase. The slopes of the stress-deflection relationship for beams with dp = 400, 450, 500, 550 and 600 mm are 1.70, 1.96, 2.45, 2.86 and 3.30 MPa/mm, respectively.   Figure 9a,b illustrate the tendon stress versus midspan moment and deflection for beams with various tendon depths, respectively. As the tendon depth increases, the ultimate tendon stress increases substantially. As mentioned previously, the beams with different tendon depths exhibit approximately the same deformation at ultimate, implying there are approximately the same concrete strain distributions along the depth of the sections of these beams. Therefore, the larger the tendon depth, the larger the average change in the concrete strain at the same level of the external tendons and, thereby, the larger the tendon stress increase. Due to a lower reduction in the effective tendon depth, a larger tendon depth assumes relatively less significant second-order effects, which would reduce relatively less tendon stresses. In this analysis, increasing the tendon depth from 400 to 600 mm causes an increase in the ultimate tendon stress increment by 93.2%. For the same deflection, the greater the tendon depth, the greater the tendon stress increase.

Effect of Tendon Elastic Modulus
The elastic modulus of CFRP tendons, Ep, ranges from 80 to 500 GPa [10]. Figure 10a,b show the effect of the tendon elastic modulus on the load-deflection and moment-curvature behavior, respectively. The beams exhibit identical behavior up to cracking. After that, the behaviors differ because the contribution of the external tendon becomes increasingly important. A higher tendon modulus of elasticity mobilizes stiffer behavior of the beams. As the tendon modulus of elasticity increases, the ultimate load and flexural strength increase, while the ultimate deflection and curvature tend to decrease. As presented in Table 4, increasing the tendon elastic modulus from 80 to 500 GPa leads to a decrease in deflection ductility by 11.1% and in curvature ductility by 31.4%.

Effect of Tendon Elastic Modulus
The elastic modulus of CFRP tendons, E p , ranges from 80 to 500 GPa [10]. Figure 10a,b show the effect of the tendon elastic modulus on the load-deflection and moment-curvature behavior, respectively. The beams exhibit identical behavior up to cracking. After that, the behaviors differ because the contribution of the external tendon becomes increasingly important. A higher tendon modulus of elasticity mobilizes stiffer behavior of the beams. As the tendon modulus of elasticity increases, the ultimate load and flexural strength increase, while the ultimate deflection and curvature tend to decrease. As presented in Table 4, increasing the tendon elastic modulus from 80 to 500 GPa leads to a decrease in deflection ductility by 11.1% and in curvature ductility by 31.4%.

Effect of Tendon Elastic Modulus
The elastic modulus of CFRP tendons, Ep, ranges from 80 to 500 GPa [10]. Figure 10a,b show the effect of the tendon elastic modulus on the load-deflection and moment-curvature behavior, respectively. The beams exhibit identical behavior up to cracking. After that, the behaviors differ because the contribution of the external tendon becomes increasingly important. A higher tendon modulus of elasticity mobilizes stiffer behavior of the beams. As the tendon modulus of elasticity increases, the ultimate load and flexural strength increase, while the ultimate deflection and curvature tend to decrease. As presented in Table 4, increasing the tendon elastic modulus from 80 to 500 GPa leads to a decrease in deflection ductility by 11.1% and in curvature ductility by 31.4%.    Figure 11a,b show the effect of the tendon elastic modulus on the tendon stress versus midspan moment and deflection, respectively. At a given moment or deflection level, a high tendon elastic modulus corresponds to a significantly higher tendon stress. The ultimate stress increase in external tendons with an elastic modulus of 500 GPa is 5.17 times that of 80 GPa. It should be noted that the stress ratio of 5.17 is smaller than the modulus ratio of 6.25, as a higher tendon elastic modulus would lead to a smaller ultimate tendon strain.
The slopes of the stress-deflection relationship for beams with E p = 80, 150, 250, 360 and 500 GPa are 1.33, 2.45, 3.98, 5.57 and 7.43 MPa/mm, respectively.  Figure 11a,b show the effect of the tendon elastic modulus on the tendon stress versus midspan moment and deflection, respectively. At a given moment or deflection level, a high tendon elastic modulus corresponds to a significantly higher tendon stress. The ultimate stress increase in external tendons with an elastic modulus of 500 GPa is 5.17 times that of 80 GPa. It should be noted that the stress ratio of 5.17 is smaller than the modulus ratio of 6.25, as a higher tendon elastic modulus would lead to a smaller ultimate tendon strain. The slopes of the stress-deflection relationship for beams with Ep = 80, 150, 250, 360 and 500 GPa are 1.33, 2.45, 3.98, 5.57 and 7.43 MPa/mm, respectively.

Method
In externally post-tensioned members, the ultimate tendon stress, σpu, depends on the whole member deformation, and therefore, its accurate predictions are complicated. The calculation of σpu is essential for the flexural strength design of externally prestressed members [27][28][29][30][31][32]. The stress σpu is commonly expressed by where Δσp is the ultimate stress increment and σpe is the effective prestress.
where ω0 is the combined reinforcing index; h is the section height; L is the span and k1 is a coefficient associated with the pattern of loading on continuous beams. For the simply supported beams considered in this study, k1 = 1.0. The ω0 is expressed by where b is the section width and As and fy are the tensile steel bar area and yield strength, respectively. Figure 12 shows the Δσp-ω0 relationships for the beams with different tendon-related variables obtained by FEA along with the JGJ 92-2016 curve. By adopting the key parameter ω0, JGJ 92-2016 takes into account the effect of the tendon area, prestress level and tendon

Method
In externally post-tensioned members, the ultimate tendon stress, σ pu , depends on the whole member deformation, and therefore, its accurate predictions are complicated. The calculation of σ pu is essential for the flexural strength design of externally prestressed members [27][28][29][30][31][32]. The stress σ pu is commonly expressed by where ∆σ p is the ultimate stress increment and σ pe is the effective prestress. JGJ 92-2016 [33] recommended the following equation for predicting ∆σ p : ∆σ p = (240 − 335ω 0 )(0.45 + 5.5h/L)k 1 (6) where ω 0 is the combined reinforcing index; h is the section height; L is the span and k 1 is a coefficient associated with the pattern of loading on continuous beams. For the simply supported beams considered in this study, k 1 = 1.0. The ω 0 is expressed by where b is the section width and A s and f y are the tensile steel bar area and yield strength, respectively. Figure 12 shows the ∆σ p -ω 0 relationships for the beams with different tendon-related variables obtained by FEA along with the JGJ 92-2016 curve. By adopting the key parameter ω 0 , JGJ 92-2016 takes into account the effect of the tendon area, prestress level and tendon depth but neglects the effect of the tendon modulus of elasticity. In addition, JGJ 92-2016 significantly underestimates the value of ∆σ p , except for the beam with E p = 80 GPa.  The numerical analysis demonstrates that the ultimate tendon stress is affected by all four tendon-related variables, i.e., the tendon area, prestress level, tendon depth and modulus of elasticity. As the parameter ω0 involves three of the tendon-related variables (i.e., Ap, dp and σpe), this parameter is adopted in a new equation to be developed herein for the prediction of Δσp. As illustrated in Figure 13, fitting to the FEA data about the Δσp-ω0 relationship of beams with various tendon areas leads to the following expression:  The numerical analysis demonstrates that the ultimate tendon stress is affected by all four tendon-related variables, i.e., the tendon area, prestress level, tendon depth and modulus of elasticity. As the parameter ω 0 involves three of the tendon-related variables (i.e., A p , d p and σ pe ), this parameter is adopted in a new equation to be developed herein for the prediction of ∆σ p . As illustrated in Figure 13, fitting to the FEA data about the ∆σ p -ω 0 relationship of beams with various tendon areas leads to the following expression:  The numerical analysis demonstrates that the ultimate tendon stress is affected by all four tendon-related variables, i.e., the tendon area, prestress level, tendon depth and modulus of elasticity. As the parameter ω0 involves three of the tendon-related variables (i.e., Ap, dp and σpe), this parameter is adopted in a new equation to be developed herein for the prediction of Δσp. As illustrated in Figure 13, fitting to the FEA data about the Δσp-ω0 relationship of beams with various tendon areas leads to the following expression:  Note that the above equation does not consider the tendon modulus of elasticity, which has been demonstrated to be crucial for ∆σ p . In order to include the effect of the tendon elastic modulus, Equation (8) is modified by introducing a coefficient as follows: where λ E is a coefficient related with the tendon elastic modulus E p . The FEA data regarding the variation in λ E against E p /E ps is presented in Figure 14, where E ps is the elastic modulus of prestressing steel, taken to be equal to 195 GPa. According to the fit curve, λ E is expressed by Note that the above equation does not consider the tendon modulus of elasticity, which has been demonstrated to be crucial for Δσp. In order to include the effect of the tendon elastic modulus, Equation (8) is modified by introducing a coefficient as follows: where λE is a coefficient related with the tendon elastic modulus Ep. The FEA data regarding the variation in λE against Ep/Eps is presented in Figure 14, where Eps is the elastic modulus of prestressing steel, taken to be equal to 195 GPa. According to the fit curve, λE is expressed by The axial equilibrium of beams is given by the following equation [34]:  The axial equilibrium of beams is given by the following equation [34]: 0.85 f ck bβ 1 c u = A p σ pu + A s f y − A s f y (11) where β 1 = 0.85; A s and f y are the compressive steel bar area and yield strength, respectively and c u is the neutral axis depth. According to Equation (11), c u is calculated by The flexural strength is determined by the following equation [34]: where d s and d s are the depths of tensile and compressive bars, respectively, and d e is the effective depth of external tendons, which is given by where R d is a reduction coefficient due to second-order effects. According to [35], for third-point loading, the value of R d is calculated from

Results
A comparison of ∆σ p and M u for beams with different tendon-related variables predicted by the simplified models and FEA is presented in Table 5 and Figures 15 and 16. It is seen that JGJ 92-2016 leads to significant underestimation in ∆σ p , and, therefore, this code underestimates the flexural strength of the beams. The predicted ∆σ p is 53.5% of the FEA data, on average, with a standard deviation of 16.2%, while the predicted M u is 88.6% of the FEA data, on average, with a standard deviation of 6.9%. The proposed simplified model shows much better predictions than JGJ 92-2016. According to the proposed model, the mean discrepancy for ∆σ p is 0.9%, with a standard deviation of 11.1%, while the mean discrepancy for M u is −1.6%, with a standard deviation of 2.1%.

Conclusions
By applying a validated FEA method, a numerical study is carried out to examine the flexural behavior of concrete beams prestressed with external CFRP tendons. Particular focus is placed on variables related to CFRP tendons, i.e., the area, prestress level, depth

Conclusions
By applying a validated FEA method, a numerical study is carried out to examine the flexural behavior of concrete beams prestressed with external CFRP tendons. Particular focus is placed on variables related to CFRP tendons, i.e., the area, prestress level, depth

Conclusions
By applying a validated FEA method, a numerical study is carried out to examine the flexural behavior of concrete beams prestressed with external CFRP tendons. Particular focus is placed on variables related to CFRP tendons, i.e., the area, prestress level, depth and elastic modulus of the tendons. An analytical model is also developed. The main conclusions are: • CFRP tendons play a crucial role in the structural performance of externally prestressed beams, including the flexural stiffness, ultimate load-carrying capacity, stress increase in external tendons, deformation and ductility. A higher tendon area, initial prestress or elastic modulus causes a lower flexural ductility. The ∆σ p decreases as the tendon area or initial prestress level increases or as the tendon depth or elastic modulus increases. • JGJ 92-2016 significantly underestimates the ultimate tendon stress, and hence, this code is over-conservative for flexural strength predictions of externally CFRP prestressed beams. The predicted ∆σ p and M u by JGJ 92-2016 are 53.5% and 88.6% of the FEA data, on average, respectively. • An equation is proposed to calculate ∆σ p , considering the influence of the tendon area, effective prestress, tendon depth and modulus of elasticity. The proposed analytical model shows excellent predictions of tendon stress and flexural strength, i.e., the mean discrepancy for ∆σ p is 0.9% with a standard deviation of 11.1%, while the mean discrepancy for M u is −1.6% with a standard deviation of 2.1%.